Over the past few months, I've noticed more and more people using other services for invitations - not necessarily instead of, but rather in addition to. I, myself, use both evite and facebook for invitations - every party I throw has an invite on both services.
Evite gets more quick responses - that makes some sense. First, Facebook bugs you about lots of stuff (wall posts, etc), so you overlook those emails much more easily. Second, Facebook invites aren't going to get lost in your inbox like Evite - there's no pressure to respond. Third, Evite doesn't include any useful information in the invitation - guests have to open up the invitation in order to know when it is.
Example: I'm planning a very large event. The Evite has 1400 people on it, and the Facebook event has 650. Within two hours, I had 100 accepts on Evite and 5 on Facebook.
Facebook gets a higher response rate over all. Why? Well, an Evite comes in - you either open it... or it trickles down in your inbox. You might see one or two reminders about an event, but it's not continuously bugging you about it. On Facebook though, it's on the main page every time you log in, poking you to respond.
Example: I recently hosted a very large event. The Evite had about 800 people on it. After numerous messages, begging people to respond, I got almost 50% of people who responded. When I didn't do that, I got about 35%. The Facebook invite had about 550 people on it. 65% responded, without any bugging.
Both have their own network effect properties. On Facebook, people discover invites for (public) events through their friends - but without their friends inviting them, because of mini-feeds. Evite guest lists, however, tend to grow through direct invitations.
So what to do? I can't use Facebook instead of Evite - not everyone is on Facebook. I like the network effect of Facebook though. So what do you do? Continue using both.